"Dual Nature of Radiation and Matter"

About

Is light a wave or a particle? The answer — both. This chapter reveals the dual nature of radiation through the photoelectric effect (light as particles) and extends the idea to matter through de Broglie's hypothesis (electrons as waves). This wave-particle duality is at the heart of quantum mechanics.

Key Concepts

25.1 Photoelectric Effect

Emission of electrons from a metal surface when light of suitable frequency falls on it.

Experimental observations:

1. Threshold frequency (${\nu }_0$): Below this, NO emission occurs — regardless of intensity
2. Maximum KE of photoelectrons depends on frequency, NOT intensity
3. Stopping potential ($V_0$) is independent of intensity
4. Photocurrent is proportional to intensity (number of photons)

Thermionic emission: Electrons gain energy from heat, not photons. (Different from photoelectric effect.)

25.2 Einstein's Photoelectric Equation

$h\nu ={\varphi }_0+K_{\text{max}}$

Where ${\varphi }_0=h{\nu }_0$ = work function (minimum energy to eject an electron).

$K_{\text{max}}=h\nu -{\varphi }_0=e{V}_0$

Graph of $V_0$ vs $\nu$:

$V_0$ vs Intensity graph: Horizontal line — stopping potential does NOT depend on intensity.

25.3 Photon Picture

• Photon energy: $E=h\nu =hc/\lambda$
• Photon momentum: $p=h/\lambda =h\nu /c$
• If wavelength is doubled → energy halves ($E\propto 1/\lambda$)
• Photons are massless, travel at speed $c$

25.4 de Broglie Wavelength

Matter also has wave nature:

$\lambda =\frac{h}{p}=\frac{h}{mv}$

For an electron accelerated through potential $V$: $\lambda =\frac{h}{\sqrt{2meV}}\approx \frac{12.27}{\sqrt{V}}\text{ A˚}$

INTEXT QUESTIONS 25.1

Q1. True or false: (a) In thermionic emission, electrons gain energy from photons. — False (from heat) (b) Maximum velocity of photoelectrons is independent of frequency. — False (depends on $\nu$) (c) There exists a frequency ${\nu }_0$ below which no photoelectric effect occurs. — True

Q2. Interpret intercepts on $V_0$ vs $\nu$ graph and calculate slope.

Ans: X-intercept = ${\nu }_0$ (threshold frequency). Y-intercept = $-{\varphi }_0/e$ (negative work function/charge). Slope = $h/e=4.14\times 10^{-15}$ V⋅s — universal for all materials.

Q3. Draw graph of stopping potential vs intensity.

Ans: Horizontal straight line — stopping potential is independent of intensity.

INTEXT QUESTIONS 25.2

Q1. Calculate the momentum of a photon of frequency $\nu$.

Ans: $p=h\nu /c=h/\lambda$.

Q2. If wavelength is doubled, how does photon energy change?

Ans: $E=hc/\lambda$. If $\lambda \to 2\lambda$, $E\to E/2$. Energy halves.

Terminal Exercise

1. What is the photoelectric effect? Describe an experiment to study it.

2. State Einstein's photoelectric equation. Explain how it accounts for the experimental observations.

3. Define: (a) threshold frequency, (b) work function, (c) stopping potential.

4. The work function of sodium is 2.3 eV. Calculate: (a) threshold frequency, (b) threshold wavelength, (c) maximum KE of photoelectrons when light of $\lambda =400$ nm is used.

5. Explain why the stopping potential is independent of light intensity.

6. State de Broglie's hypothesis. Derive the expression for de Broglie wavelength.

7. Calculate the de Broglie wavelength of an electron accelerated through 100 V.

8. An electron and a proton have the same kinetic energy. Which has the larger de Broglie wavelength? Why?

9. Why is the wave nature of matter not apparent in our daily experience?

10. Light of frequency $1.5\times 10^{15}$ Hz falls on a metal of work function 2 eV. Find: (a) maximum KE, (b) stopping potential.

Quick Revision